London insisted there was "no prospect" of a Brexit deal unless the backstop was abandoned, after Brussels said Britain had still not come up with a workable alternative.
Johnson wrote to EU Council President Donald Tusk on Monday to insist that Britain could not accept what he called the "anti-democratic" backstop, a mechanism to avoid border checks between EU-member Ireland and Northern Ireland, part of the UK.
Since taking office last month, Johnson has been adamant Britain will leave the European Union on October 31 come what may and has stepped up preparations for a "no deal" departure that would cause major economic disruption.
But the European Commission, the EU executive which has led Brexit negotiations with London, dismissed the proposal in Johnson's letter that the backstop could be replaced with a "commitment" to find "alternative arrangements".
"The letter does not provide a legal operational solution to prevent the return of a hard border on the island of Ireland," commission spokeswoman Natasha Bertaud told reporters.
"It does not set out what any alternative arrangements could be, and in fact it recognises there is no guarantee that such arrangements will be in place by the end of the transitional period."
- 'No prospect' of deal with backstop -
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Downing Street hit back, saying the withdrawal agreement struck in 2018, which includes the backstop, was "simply unviable", having been rejected three times by British lawmakers.
"Unless the withdrawal agreement is reopened and the backstop abolished there is no prospect of a deal," a Downing Street spokesman said.
The clash comes as Johnson prepares to travel to Berlin and Paris, where he hopes to convince German Chancellor Angela Merkel and French President Emmanuel Macron to give ground in the search for a Brexit deal.
Telephone talks of nearly an hour on Monday with Irish Prime Minister Leo Varadkar failed to yield any obvious breakthrough.
Johnson's diplomatic offensive continues at a G7 summit in the French resort of Biarritz at the weekend, where Johnson hopes to show off his warm relations with US President Donald Trump as a signal of post-Brexit Britain's global ambitions.
Johnson said Britain's European partners were being "a bit negative", with Tusk's approach hardly "redolent of a sense of optimism", he told reporters.
"But I think, actually, we'll get there. There is a real sense now that something needs to be done with this backstop."
He said the EU would not make concessions so long as it thought parliament might block Brexit, "so it's going to take a bit of patience".
In another move designed to signal the seriousness of Britain's intent to leave the bloc on October 31, the government announced that British officials will stop attending most EU meetings from September 1.
Officials will only go to meetings on "matters of ongoing national interest" such as security, the Brexit ministry announced, saying the time saved could be used to focus on other "national priorities".
- Gored by Tusk -
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Brussels insists that the backstop -- which would keep the UK in EU customs arrangements to prevent a hard border on the island of Ireland -- is essential to preserve the integrity of European trade and to avoid risking a return of sectarian violence in Northern Ireland.
Tusk took to Twitter on Tuesday to give a robust response to Johnson.
"The backstop is an insurance to avoid a hard border on the island of Ireland unless and until an alternative is found," Tusk wrote.
"Those against the backstop and not proposing realistic alternatives in fact support reestablishing a border. Even if they do not admit it."
Critics say the backstop would keep Britain tied to EU rules potentially indefinitely.
But the EU and Ireland say that British offers to find "alternative arrangements" are too vague and give no solid legal guarantees, amounting to little more than an invitation to trust London.
Brussels has repeatedly said it will not reopen or renegotiate the near 600-page withdrawal agreement, but is willing to tweak the accompanying "political declaration" on future EU-UK ties.
So far the UK has not requested any meetings, commission spokeswoman Bertaud said -- which suggests London is pinning hopes for progress on the one-on-one meetings with Merkel on Wednesday and Macron on Thursday.
With both sides appearing intransigent, fears are growing that Britain will crash out without a deal.